Calculus, 11 9 #4, Power Series Representation

TL;DR

The video explains how to write a power series expansion for the function 5 over 1 minus 4x squared.

Transcript

we're going to write a power series expansion for the function 5 over 1 minus 4x squared and to do that we just have to match this with our best friend here we have the one on the top but the function here we have is the 5. let's put a 5 on the side and then as we can see these two ones they match that's good so it's the minus sign so i can just sa... Read More

Key Insights

• ✊ Power series expansions are useful for approximating functions.
• 😑 Matching terms and using sigma notation can make the power series expression more compact.
• ☺️ The radius of convergence determines the range of x values for which the power series converges.
• ☺️ The interval of convergence is the range of x values that make the series convergent.

Questions & Answers

Q: How do you write a power series expansion for the function 5 over 1 minus 4x squared?

To write the power series expansion, you need to match the terms by putting a 5 on the side and using sigma notation to express the series. The expansion will involve raising 4x squared to different powers in each term.

Q: What is the radius of convergence for the power series?

The radius of convergence can be found by isolating the absolute value of x in the inequality |x|^2 < 1/4. Solving this inequality gives the radius of convergence as 1/2.

Q: What is the interval of convergence for the power series?

The interval of convergence can be determined by considering the center and radius. In this case, the center is 0 and the radius is 1/2. Therefore, the interval of convergence is (-1/2, 1/2).

Q: Do we need to check for convergence at the endpoints?

No, we don't need to check for convergence at the endpoints in this case because the power series was obtained using algebraic manipulation without differentiation or integration. The endpoints are not included in the interval of convergence.

Summary & Key Takeaways

• The video teaches how to write a power series expansion for the function 5 over 1 minus 4x squared.

• The process involves matching the terms and using sigma notation to express the series more compactly.

• The video also discusses the radius of convergence and interval of convergence for the power series.